Scalar 1-loop Feynman integrals as meromorphic functions in space-time dimension d

Khiem Hong Phan,Tord Riemann

doi:10.1016/j.physletb.2019.02.044

Khiem Hong Phan, Tord Riemann

Open Access

https://doi.org/10.1016/j.physletb.2019.02.044

Copy DOI

Abstract

The long-standing problem of representing the general massive one-loop Feynman integral as a meromorphic function of the space-time dimension d has been solved for the basis of scalar one- to four-point functions with indices one. In 2003 the solution of difference equations in the space-time dimension allowed to determine the necessary classes of special functions: self-energies need ordinary logarithms and Gauss hypergeometric functions F12, vertices need additionally Kampé de Fériet-Appell functions F1, and box integrals also Lauricella-Saran functions FS. In this study, alternative recursive Mellin-Barnes representations are used for the representation of n-point functions in terms of (n−1)-point functions. The approach enabled the first derivation of explicit solutions for the Feynman integrals at arbitrary kinematics. In this article, we scetch our new representations for the general massive vertex and box Feynman integrals and derive a numerical approach for the necessary Appell functions F1 and Saran functions FS at arbitrary kinematical arguments.

Highlights

We are studying scalar one-loop Feynman integrals, Jn(d) = ddk iπd/2 D1ν1 1 D2ν2 · · Dnνn (1)with inverse propagators Di = (k + qi)2 − m2i + iε
Dimensions d = 4 + 2n − 2ǫ with n ≥ 0 are of physical interest because tensor one-loop Feynman integrals of rank r in 4 − 2ǫ dimensions may be expressed by scalar integrals taken in higher dimensions up to d = 4 + 2r − 2ǫ
A systematic numerical treatment of the terms of order ǫ-terms was performed in 1992 [5], and a systematic numerical approach was worked out in 2001 [6]. It has been shown in 2003 [7, 8] that representations in general dimension d, including d = 4 − 2ǫ

Summary

Introduction

A scetch of the Feynman integrals at arbitrary kinematics in terms of 2F1, F1, FS and their explicit numerical determination are the subject of this letter. Representations of the massive self-energy, vertex and box integrals can be derived iteratively from (10) by closing the integration contours of the Mellin-Barnes integrals e.g. to the right and taking the two series of residues of the corresponding Γ-functions with arguments (−s + · · · ). There, an infinite sum over a discrete dimensional parameter s was derived in order to represent an n-point function Jn(d) by integrals Jn−1(d + 2s). The term of J123 in (15) with d-independent F1 and FS It replaces the so-called b3-term of the vertex integral in [8] for arbitrary kinematics, while the d-dimensional parts of J1234 agree. In Appendix A to Appendix C we will show how to calculate the various F1 and FS for arbitrary complex arguments; for 2F1 we assume that such calculations are well-known

Numerical results

Discussion